TPTP Problem File: ITP105^1.p
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%------------------------------------------------------------------------------
% File : ITP105^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListSlice problem prob_61__5615344_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : ListSlice/prob_61__5615344_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 444 ( 223 unt; 94 typ; 0 def)
% Number of atoms : 887 ( 375 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 2086 ( 85 ~; 26 |; 46 &;1677 @)
% ( 0 <=>; 252 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 210 ( 210 >; 0 *; 0 +; 0 <<)
% Number of symbols : 84 ( 81 usr; 10 con; 0-3 aty)
% Number of variables : 646 ( 42 ^; 582 !; 22 ?; 646 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:29:59.697
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
list_l2071841302list_a: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
option_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_Itf__a_J_J,type,
option_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (81)
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
inverse_inverse_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001t__List__Olist_Itf__a_J,type,
listSl162220270list_a: list_list_a > nat > list_list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice_001tf__a,type,
listSl97544552lice_a: list_a > nat > list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001t__List__Olist_Itf__a_J,type,
listSl856612276list_a: list_list_a > nat > nat > list_list_list_a ).
thf(sy_c_ListSlice__Mirabelle__csprijwpmv_Olist__slice__aux_001tf__a,type,
listSl1495374126_aux_a: list_a > nat > nat > list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bind_l311701255list_a: list_list_list_a > ( list_list_a > list_list_list_a ) > list_list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
bind_l94945665list_a: list_list_list_a > ( list_list_a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
bind_list_list_a_a: list_list_list_a > ( list_list_a > list_a ) > list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bind_l971853709list_a: list_list_a > ( list_a > list_list_list_a ) > list_list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
bind_list_a_list_a: list_list_a > ( list_a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001t__List__Olist_Itf__a_J_001tf__a,type,
bind_list_a_a: list_list_a > ( list_a > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bind_a_list_list_a: list_a > ( a > list_list_list_a ) > list_list_list_a ).
thf(sy_c_List_Obind_001tf__a_001t__List__Olist_Itf__a_J,type,
bind_a_list_a: list_a > ( a > list_list_a ) > list_list_a ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
concat_list_list_a: list_l2071841302list_a > list_list_list_a ).
thf(sy_c_List_Oconcat_001t__List__Olist_Itf__a_J,type,
concat_list_a: list_list_list_a > list_list_a ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
count_429956358list_a: list_list_list_a > list_list_a > nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_Itf__a_J,type,
count_list_list_a: list_list_a > list_a > nat ).
thf(sy_c_List_Ocount__list_001tf__a,type,
count_list_a: list_a > a > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
gen_le840878749list_a: nat > list_list_list_a > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_Itf__a_J,type,
gen_length_list_a: nat > list_list_a > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
nil_list_list_list_a: list_l2071841302list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_ex1_list_list_a: ( list_list_a > $o ) > list_list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Omap__filter_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
map_fi824137674list_a: ( list_list_a > option_list_list_a ) > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Omap__filter_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
map_fi213215172list_a: ( list_list_a > option_list_a ) > list_list_list_a > list_list_a ).
thf(sy_c_List_Omap__filter_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
map_fi1574988606st_a_a: ( list_list_a > option_a ) > list_list_list_a > list_a ).
thf(sy_c_List_Omap__filter_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
map_fi1090123216list_a: ( list_a > option_list_list_a ) > list_list_a > list_list_list_a ).
thf(sy_c_List_Omap__filter_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_fi461586634list_a: ( list_a > option_list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Omap__filter_001t__List__Olist_Itf__a_J_001tf__a,type,
map_filter_list_a_a: ( list_a > option_a ) > list_list_a > list_a ).
thf(sy_c_List_Omap__filter_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
map_fi2089402454list_a: ( a > option_list_list_a ) > list_a > list_list_list_a ).
thf(sy_c_List_Omap__filter_001tf__a_001t__List__Olist_Itf__a_J,type,
map_filter_a_list_a: ( a > option_list_a ) > list_a > list_list_a ).
thf(sy_c_List_Omap__filter_001tf__a_001tf__a,type,
map_filter_a_a: ( a > option_a ) > list_a > list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
maps_l57963201list_a: ( list_list_a > list_list_list_a ) > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
maps_l2025957947list_a: ( list_list_a > list_list_a ) > list_list_list_a > list_list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
maps_list_list_a_a: ( list_list_a > list_a ) > list_list_list_a > list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
maps_l755382343list_a: ( list_a > list_list_list_a ) > list_list_a > list_list_list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
maps_list_a_list_a: ( list_a > list_list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Omaps_001t__List__Olist_Itf__a_J_001tf__a,type,
maps_list_a_a: ( list_a > list_a ) > list_list_a > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
maps_a_list_list_a: ( a > list_list_list_a ) > list_a > list_list_list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__List__Olist_Itf__a_J,type,
maps_a_list_a: ( a > list_list_a ) > list_a > list_list_a ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_Omember_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_list_a > list_list_a > $o ).
thf(sy_c_List_Omember_001t__List__Olist_Itf__a_J,type,
member_list_a: list_list_a > list_a > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_Onull_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
null_list_list_a: list_list_list_a > $o ).
thf(sy_c_List_Onull_001t__List__Olist_Itf__a_J,type,
null_list_a: list_list_a > $o ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri2019852685at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1382578993at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri2110766477t_real: nat > real ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
size_s575106428list_a: list_list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s1427607542list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_k,type,
k: nat ).
% Relevant facts (346)
thf(fact_0_list__slice__0,axiom,
! [Xs: list_list_a] :
( ( listSl162220270list_a @ Xs @ zero_zero_nat )
= nil_list_list_a ) ).
% list_slice_0
thf(fact_1_list__slice__0,axiom,
! [Xs: list_a] :
( ( listSl97544552lice_a @ Xs @ zero_zero_nat )
= nil_list_a ) ).
% list_slice_0
thf(fact_2_concat_Osimps_I1_J,axiom,
( ( concat_list_list_a @ nil_list_list_list_a )
= nil_list_list_a ) ).
% concat.simps(1)
thf(fact_3_concat_Osimps_I1_J,axiom,
( ( concat_list_a @ nil_list_list_a )
= nil_list_a ) ).
% concat.simps(1)
thf(fact_4_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_5_list__ex1__simps_I1_J,axiom,
! [P: list_list_a > $o] :
~ ( list_ex1_list_list_a @ P @ nil_list_list_a ) ).
% list_ex1_simps(1)
thf(fact_6_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_7_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_8_list__slice__less,axiom,
! [Xs: list_list_a,K: nat] :
( ( ord_less_nat @ ( size_s1427607542list_a @ Xs ) @ K )
=> ( ( listSl162220270list_a @ Xs @ K )
= nil_list_list_a ) ) ).
% list_slice_less
thf(fact_9_list__slice__less,axiom,
! [Xs: list_a,K: nat] :
( ( ord_less_nat @ ( size_size_list_a @ Xs ) @ K )
=> ( ( listSl97544552lice_a @ Xs @ K )
= nil_list_a ) ) ).
% list_slice_less
thf(fact_10_bind__simps_I1_J,axiom,
! [F: a > list_list_list_a] :
( ( bind_a_list_list_a @ nil_a @ F )
= nil_list_list_a ) ).
% bind_simps(1)
thf(fact_11_bind__simps_I1_J,axiom,
! [F: list_a > list_list_list_a] :
( ( bind_l971853709list_a @ nil_list_a @ F )
= nil_list_list_a ) ).
% bind_simps(1)
thf(fact_12_bind__simps_I1_J,axiom,
! [F: list_list_a > list_a] :
( ( bind_list_list_a_a @ nil_list_list_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_13_bind__simps_I1_J,axiom,
! [F: list_list_a > list_list_a] :
( ( bind_l94945665list_a @ nil_list_list_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_14_bind__simps_I1_J,axiom,
! [F: list_list_a > list_list_list_a] :
( ( bind_l311701255list_a @ nil_list_list_a @ F )
= nil_list_list_a ) ).
% bind_simps(1)
thf(fact_15_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_16_bind__simps_I1_J,axiom,
! [F: a > list_list_a] :
( ( bind_a_list_a @ nil_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_17_bind__simps_I1_J,axiom,
! [F: list_a > list_a] :
( ( bind_list_a_a @ nil_list_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_18_bind__simps_I1_J,axiom,
! [F: list_a > list_list_a] :
( ( bind_list_a_list_a @ nil_list_a @ F )
= nil_list_a ) ).
% bind_simps(1)
thf(fact_19_member__rec_I2_J,axiom,
! [Y: list_list_a] :
~ ( member_list_list_a @ nil_list_list_a @ Y ) ).
% member_rec(2)
thf(fact_20_member__rec_I2_J,axiom,
! [Y: a] :
~ ( member_a @ nil_a @ Y ) ).
% member_rec(2)
thf(fact_21_member__rec_I2_J,axiom,
! [Y: list_a] :
~ ( member_list_a @ nil_list_a @ Y ) ).
% member_rec(2)
thf(fact_22_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le840878749list_a @ N @ nil_list_list_a )
= N ) ).
% gen_length_code(1)
thf(fact_23_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_24_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_a @ N @ nil_list_a )
= N ) ).
% gen_length_code(1)
thf(fact_25_maps__simps_I2_J,axiom,
! [F: a > list_list_list_a] :
( ( maps_a_list_list_a @ F @ nil_a )
= nil_list_list_a ) ).
% maps_simps(2)
thf(fact_26_maps__simps_I2_J,axiom,
! [F: list_a > list_list_list_a] :
( ( maps_l755382343list_a @ F @ nil_list_a )
= nil_list_list_a ) ).
% maps_simps(2)
thf(fact_27_maps__simps_I2_J,axiom,
! [F: list_list_a > list_a] :
( ( maps_list_list_a_a @ F @ nil_list_list_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_28_maps__simps_I2_J,axiom,
! [F: list_list_a > list_list_a] :
( ( maps_l2025957947list_a @ F @ nil_list_list_a )
= nil_list_a ) ).
% maps_simps(2)
thf(fact_29_maps__simps_I2_J,axiom,
! [F: list_list_a > list_list_list_a] :
( ( maps_l57963201list_a @ F @ nil_list_list_a )
= nil_list_list_a ) ).
% maps_simps(2)
thf(fact_30_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_31_maps__simps_I2_J,axiom,
! [F: a > list_list_a] :
( ( maps_a_list_a @ F @ nil_a )
= nil_list_a ) ).
% maps_simps(2)
thf(fact_32_maps__simps_I2_J,axiom,
! [F: list_a > list_a] :
( ( maps_list_a_a @ F @ nil_list_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_33_maps__simps_I2_J,axiom,
! [F: list_a > list_list_a] :
( ( maps_list_a_list_a @ F @ nil_list_a )
= nil_list_a ) ).
% maps_simps(2)
thf(fact_34_null__rec_I2_J,axiom,
null_list_list_a @ nil_list_list_a ).
% null_rec(2)
thf(fact_35_null__rec_I2_J,axiom,
null_a @ nil_a ).
% null_rec(2)
thf(fact_36_null__rec_I2_J,axiom,
null_list_a @ nil_list_a ).
% null_rec(2)
thf(fact_37_eq__Nil__null,axiom,
! [Xs: list_a] :
( ( Xs = nil_a )
= ( null_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_38_eq__Nil__null,axiom,
! [Xs: list_list_a] :
( ( Xs = nil_list_a )
= ( null_list_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_39_eq__Nil__null,axiom,
! [Xs: list_list_list_a] :
( ( Xs = nil_list_list_a )
= ( null_list_list_a @ Xs ) ) ).
% eq_Nil_null
thf(fact_40_map__filter__simps_I2_J,axiom,
! [F: a > option_a] :
( ( map_filter_a_a @ F @ nil_a )
= nil_a ) ).
% map_filter_simps(2)
thf(fact_41_map__filter__simps_I2_J,axiom,
! [F: a > option_list_a] :
( ( map_filter_a_list_a @ F @ nil_a )
= nil_list_a ) ).
% map_filter_simps(2)
thf(fact_42_map__filter__simps_I2_J,axiom,
! [F: a > option_list_list_a] :
( ( map_fi2089402454list_a @ F @ nil_a )
= nil_list_list_a ) ).
% map_filter_simps(2)
thf(fact_43_map__filter__simps_I2_J,axiom,
! [F: list_a > option_a] :
( ( map_filter_list_a_a @ F @ nil_list_a )
= nil_a ) ).
% map_filter_simps(2)
thf(fact_44_map__filter__simps_I2_J,axiom,
! [F: list_a > option_list_a] :
( ( map_fi461586634list_a @ F @ nil_list_a )
= nil_list_a ) ).
% map_filter_simps(2)
thf(fact_45_map__filter__simps_I2_J,axiom,
! [F: list_a > option_list_list_a] :
( ( map_fi1090123216list_a @ F @ nil_list_a )
= nil_list_list_a ) ).
% map_filter_simps(2)
thf(fact_46_map__filter__simps_I2_J,axiom,
! [F: list_list_a > option_a] :
( ( map_fi1574988606st_a_a @ F @ nil_list_list_a )
= nil_a ) ).
% map_filter_simps(2)
thf(fact_47_map__filter__simps_I2_J,axiom,
! [F: list_list_a > option_list_a] :
( ( map_fi213215172list_a @ F @ nil_list_list_a )
= nil_list_a ) ).
% map_filter_simps(2)
thf(fact_48_map__filter__simps_I2_J,axiom,
! [F: list_list_a > option_list_list_a] :
( ( map_fi824137674list_a @ F @ nil_list_list_a )
= nil_list_list_a ) ).
% map_filter_simps(2)
thf(fact_49_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s1427607542list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_50_length__0__conv,axiom,
! [Xs: list_list_list_a] :
( ( ( size_s575106428list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_list_a ) ) ).
% length_0_conv
thf(fact_51_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_52_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1427607542list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_53_length__greater__0__conv,axiom,
! [Xs: list_list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s575106428list_a @ Xs ) )
= ( Xs != nil_list_list_a ) ) ).
% length_greater_0_conv
thf(fact_54_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_55_list_Osize_I3_J,axiom,
( ( size_s1427607542list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_56_list_Osize_I3_J,axiom,
( ( size_s575106428list_a @ nil_list_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_57_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_58_length__code,axiom,
( size_s1427607542list_a
= ( gen_length_list_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_59_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_60_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_61_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_62_neq__if__length__neq,axiom,
! [Xs: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_63_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_64_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_65_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_66_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_67_length__greater__imp__not__empty,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s1427607542list_a @ Xs ) )
=> ( Xs != nil_list_a ) ) ).
% length_greater_imp_not_empty
thf(fact_68_length__greater__imp__not__empty,axiom,
! [N: nat,Xs: list_list_list_a] :
( ( ord_less_nat @ N @ ( size_s575106428list_a @ Xs ) )
=> ( Xs != nil_list_list_a ) ) ).
% length_greater_imp_not_empty
thf(fact_69_length__greater__imp__not__empty,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( Xs != nil_a ) ) ).
% length_greater_imp_not_empty
thf(fact_70_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_71_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_72_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_73_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_74_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_75_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_76_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_77_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_78_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_79_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_80_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_81_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_82_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_83_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_84_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_85_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_86_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_87_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_88_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A2: set_real] :
( ( collect_real
@ ^ [X2: real] : ( member_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_92_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_93_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_94_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_95_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_96_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_97_gr__implies__gr0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_implies_gr0
thf(fact_98_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_99_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_100_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_101_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_102_list__slice__aux_Osimps_I1_J,axiom,
! [Xs: list_a,K: nat] :
( ( listSl1495374126_aux_a @ Xs @ K @ zero_zero_nat )
= nil_list_a ) ).
% list_slice_aux.simps(1)
thf(fact_103_list__slice__aux_Osimps_I1_J,axiom,
! [Xs: list_list_a,K: nat] :
( ( listSl856612276list_a @ Xs @ K @ zero_zero_nat )
= nil_list_list_a ) ).
% list_slice_aux.simps(1)
thf(fact_104_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_105_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_106_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri2110766477t_real @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_107_count__list_Osimps_I1_J,axiom,
! [Y: a] :
( ( count_list_a @ nil_a @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_108_count__list_Osimps_I1_J,axiom,
! [Y: list_a] :
( ( count_list_list_a @ nil_list_a @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_109_count__list_Osimps_I1_J,axiom,
! [Y: list_list_a] :
( ( count_429956358list_a @ nil_list_list_a @ Y )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_110_interval__induct__rule,axiom,
! [I: set_nat,P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y2: nat] :
( ( ( member_nat @ X3 @ I )
& ( member_nat @ Y2 @ I )
& ( ord_less_nat @ Y2 @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% interval_induct_rule
thf(fact_111_interval__induct,axiom,
! [I: set_nat,P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y2: nat] :
( ( ( member_nat @ X3 @ I )
& ( member_nat @ Y2 @ I )
& ( ord_less_nat @ Y2 @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% interval_induct
thf(fact_112_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_113_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri2019852685at_int @ M )
= ( semiri2019852685at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_114_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri2110766477t_real @ M )
= ( semiri2110766477t_real @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_115_of__nat__0,axiom,
( ( semiri1382578993at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_116_of__nat__0,axiom,
( ( semiri2019852685at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_117_of__nat__0,axiom,
( ( semiri2110766477t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_118_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1382578993at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_119_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri2019852685at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_120_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_real
= ( semiri2110766477t_real @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_121_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1382578993at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_122_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri2019852685at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_123_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri2110766477t_real @ M )
= zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_124_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_125_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_126_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_127_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_128_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_129_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_real @ ( semiri2110766477t_real @ M ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_130_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_131_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_132_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_133_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_134_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_135_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_136_Collect__minI,axiom,
! [K: nat,I: set_nat,P: nat > $o] :
( ( member_nat @ K @ I )
=> ( ( P @ K )
=> ? [X3: nat] :
( ( member_nat @ X3 @ I )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ I )
=> ( ( ord_less_nat @ Xa @ X3 )
=> ~ ( P @ Xa ) ) ) ) ) ) ).
% Collect_minI
thf(fact_137_Collect__minI__ex,axiom,
! [I: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ I )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ I )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ I )
=> ( ( ord_less_nat @ Xa @ X3 )
=> ~ ( P @ Xa ) ) ) ) ) ).
% Collect_minI_ex
thf(fact_138_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri2019852685at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_139_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri2019852685at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_140_reals__Archimedean2,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_real @ X @ ( semiri2110766477t_real @ N2 ) ) ).
% reals_Archimedean2
thf(fact_141_list__slice__def,axiom,
( listSl97544552lice_a
= ( ^ [Xs3: list_a,K2: nat] : ( listSl1495374126_aux_a @ Xs3 @ K2 @ ( divide_divide_nat @ ( size_size_list_a @ Xs3 ) @ K2 ) ) ) ) ).
% list_slice_def
thf(fact_142_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_143_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri2019852685at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_144_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri2110766477t_real @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_145_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_146_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_147_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ zero_zero_real )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_148_ex__inverse__of__nat__less,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N2 ) ) @ X ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_149_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_150_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_151_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_152_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_153_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_154_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_real @ ( semiri2110766477t_real @ M ) @ ( semiri2110766477t_real @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_155_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_156_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri1382578993at_nat @ ( power_power_nat @ M @ N ) )
= ( power_power_nat @ ( semiri1382578993at_nat @ M ) @ N ) ) ).
% of_nat_power
thf(fact_157_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri2019852685at_int @ ( power_power_nat @ M @ N ) )
= ( power_power_int @ ( semiri2019852685at_int @ M ) @ N ) ) ).
% of_nat_power
thf(fact_158_of__nat__power,axiom,
! [M: nat,N: nat] :
( ( semiri2110766477t_real @ ( power_power_nat @ M @ N ) )
= ( power_power_real @ ( semiri2110766477t_real @ M ) @ N ) ) ).
% of_nat_power
thf(fact_159_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W )
= ( semiri1382578993at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_160_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri2019852685at_int @ B ) @ W )
= ( semiri2019852685at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_161_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_real @ ( semiri2110766477t_real @ B ) @ W )
= ( semiri2110766477t_real @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_162_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1382578993at_nat @ X )
= ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_163_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri2019852685at_int @ X )
= ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_164_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri2110766477t_real @ X )
= ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_165_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_166_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_167_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_real @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) @ ( semiri2110766477t_real @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_168_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_169_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_170_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_real @ ( semiri2110766477t_real @ X ) @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_171_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_172_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_173_power__eq__0__iff,axiom,
! [A: real,N: nat] :
( ( ( power_power_real @ A @ N )
= zero_zero_real )
= ( ( A = zero_zero_real )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_174_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1382578993at_nat @ X ) @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_175_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri2019852685at_int @ X ) @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_176_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_real @ ( semiri2110766477t_real @ X ) @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_177_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1382578993at_nat @ B ) @ W ) @ ( semiri1382578993at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_178_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri2019852685at_int @ B ) @ W ) @ ( semiri2019852685at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_179_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_real @ ( power_power_real @ ( semiri2110766477t_real @ B ) @ W ) @ ( semiri2110766477t_real @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_180_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_181_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_182_power__mono__iff,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
= ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_183_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri2019852685at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_184_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri2019852685at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_185_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_186_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1382578993at_nat @ I2 ) @ ( semiri1382578993at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_187_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri2019852685at_int @ I2 ) @ ( semiri2019852685at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_188_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_real @ ( semiri2110766477t_real @ I2 ) @ ( semiri2110766477t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_189_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_190_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri2019852685at_int @ M )
= ( semiri2019852685at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_191_real__arch__simple,axiom,
! [X: real] :
? [N2: nat] : ( ord_less_eq_real @ X @ ( semiri2110766477t_real @ N2 ) ) ).
% real_arch_simple
thf(fact_192_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_193_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_194_power__less__imp__less__base,axiom,
! [A: real,N: nat,B: real] :
( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_195_power__inverse,axiom,
! [A: real,N: nat] :
( ( power_power_real @ ( inverse_inverse_real @ A ) @ N )
= ( inverse_inverse_real @ ( power_power_real @ A @ N ) ) ) ).
% power_inverse
thf(fact_196_power__divide,axiom,
! [A: real,B: real,N: nat] :
( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
= ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).
% power_divide
thf(fact_197_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_198_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_199_power__not__zero,axiom,
! [A: real,N: nat] :
( ( A != zero_zero_real )
=> ( ( power_power_real @ A @ N )
!= zero_zero_real ) ) ).
% power_not_zero
thf(fact_200_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_201_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_202_zero__le__power,axiom,
! [A: real,N: nat] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_le_power
thf(fact_203_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_204_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_205_power__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).
% power_mono
thf(fact_206_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_207_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_208_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_209_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_210_power__eq__imp__eq__base,axiom,
! [A: real,N: nat,B: real] :
( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_211_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_212_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_213_power__eq__iff__eq__base,axiom,
! [N: nat,A: real,B: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ( ( power_power_real @ A @ N )
= ( power_power_real @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_214_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
= ( ? [N3: nat] :
( ( N3 != zero_zero_nat )
& ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N3 ) ) )
& ( ord_less_real @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N3 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_215_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D: real,E: real] :
( ( ord_less_real @ D @ E )
=> ( ( P @ D )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( P @ ( inverse_inverse_real @ ( semiri2110766477t_real @ N2 ) ) ) )
=> ( ( ord_less_real @ zero_zero_real @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_216_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_217_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_218_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_219_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_220_ge__less__neq__conv,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,N3: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ A3 )
=> ( N3 != X2 ) ) ) ) ).
% ge_less_neq_conv
thf(fact_221_ge__less__neq__conv,axiom,
( ord_less_eq_int
= ( ^ [A3: int,N3: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ A3 )
=> ( N3 != X2 ) ) ) ) ).
% ge_less_neq_conv
thf(fact_222_ge__less__neq__conv,axiom,
( ord_less_eq_real
= ( ^ [A3: real,N3: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ A3 )
=> ( N3 != X2 ) ) ) ) ).
% ge_less_neq_conv
thf(fact_223_less__ge__neq__conv,axiom,
( ord_less_nat
= ( ^ [N3: nat,A3: nat] :
! [X2: nat] :
( ( ord_less_eq_nat @ A3 @ X2 )
=> ( N3 != X2 ) ) ) ) ).
% less_ge_neq_conv
thf(fact_224_less__ge__neq__conv,axiom,
( ord_less_int
= ( ^ [N3: int,A3: int] :
! [X2: int] :
( ( ord_less_eq_int @ A3 @ X2 )
=> ( N3 != X2 ) ) ) ) ).
% less_ge_neq_conv
thf(fact_225_less__ge__neq__conv,axiom,
( ord_less_real
= ( ^ [N3: real,A3: real] :
! [X2: real] :
( ( ord_less_eq_real @ A3 @ X2 )
=> ( N3 != X2 ) ) ) ) ).
% less_ge_neq_conv
thf(fact_226_greater__le__neq__conv,axiom,
( ord_less_nat
= ( ^ [A3: nat,N3: nat] :
! [X2: nat] :
( ( ord_less_eq_nat @ X2 @ A3 )
=> ( N3 != X2 ) ) ) ) ).
% greater_le_neq_conv
thf(fact_227_greater__le__neq__conv,axiom,
( ord_less_int
= ( ^ [A3: int,N3: int] :
! [X2: int] :
( ( ord_less_eq_int @ X2 @ A3 )
=> ( N3 != X2 ) ) ) ) ).
% greater_le_neq_conv
thf(fact_228_greater__le__neq__conv,axiom,
( ord_less_real
= ( ^ [A3: real,N3: real] :
! [X2: real] :
( ( ord_less_eq_real @ X2 @ A3 )
=> ( N3 != X2 ) ) ) ) ).
% greater_le_neq_conv
thf(fact_229_le__greater__neq__conv,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,A3: nat] :
! [X2: nat] :
( ( ord_less_nat @ A3 @ X2 )
=> ( N3 != X2 ) ) ) ) ).
% le_greater_neq_conv
thf(fact_230_le__greater__neq__conv,axiom,
( ord_less_eq_int
= ( ^ [N3: int,A3: int] :
! [X2: int] :
( ( ord_less_int @ A3 @ X2 )
=> ( N3 != X2 ) ) ) ) ).
% le_greater_neq_conv
thf(fact_231_le__greater__neq__conv,axiom,
( ord_less_eq_real
= ( ^ [N3: real,A3: real] :
! [X2: real] :
( ( ord_less_real @ A3 @ X2 )
=> ( N3 != X2 ) ) ) ) ).
% le_greater_neq_conv
thf(fact_232_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_233_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_234_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_235_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_236_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_237_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_238_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_239_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_240_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_241_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_242_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_243_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_244_zero__less__power,axiom,
! [A: real,N: nat] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N ) ) ) ).
% zero_less_power
thf(fact_245_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_246_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_247_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_248_power__strict__mono,axiom,
! [A: real,B: real,N: nat] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_249_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_250_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_251_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_real @ zero_zero_real @ N )
= zero_zero_real ) ) ).
% zero_power
thf(fact_252_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_253_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_254_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2110766477t_real @ N ) ) ).
% of_nat_0_le_iff
thf(fact_255_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_256_count__le__length,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( size_size_list_a @ Xs ) ) ).
% count_le_length
thf(fact_257_list__slice__length,axiom,
! [Xs: list_a,K: nat] :
( ( size_s1427607542list_a @ ( listSl97544552lice_a @ Xs @ K ) )
= ( divide_divide_nat @ ( size_size_list_a @ Xs ) @ K ) ) ).
% list_slice_length
thf(fact_258_inverse__le__iff__le,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le
thf(fact_259_inverse__le__iff__le__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_eq_real @ B @ A ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_260_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( divide_divide_nat @ M @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_261_inverse__positive__iff__positive,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% inverse_positive_iff_positive
thf(fact_262_inverse__negative__iff__negative,axiom,
! [A: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% inverse_negative_iff_negative
thf(fact_263_inverse__less__iff__less__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_264_inverse__eq__iff__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_265_inverse__inverse__eq,axiom,
! [A: real] :
( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_266_division__ring__divide__zero,axiom,
! [A: real] :
( ( divide_divide_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% division_ring_divide_zero
thf(fact_267_divide__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ( divide_divide_real @ A @ C )
= ( divide_divide_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_268_divide__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ( divide_divide_real @ C @ A )
= ( divide_divide_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_269_divide__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( divide_divide_real @ A @ B )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divide_eq_0_iff
thf(fact_270_inverse__nonzero__iff__nonzero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_271_inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% inverse_zero
thf(fact_272_inverse__divide,axiom,
! [A: real,B: real] :
( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
= ( divide_divide_real @ B @ A ) ) ).
% inverse_divide
thf(fact_273_inverse__nonnegative__iff__nonnegative,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_274_inverse__nonpositive__iff__nonpositive,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_275_inverse__less__iff__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
= ( ord_less_real @ B @ A ) ) ) ) ).
% inverse_less_iff_less
thf(fact_276_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_277_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_278_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y3: real] :
( ( ord_less_real @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_279_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_280_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_281_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_282_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_283_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_284_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_285_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq_real @ ( semiri2110766477t_real @ ( divide_divide_nat @ N @ X ) ) @ ( divide_divide_real @ ( semiri2110766477t_real @ N ) @ ( semiri2110766477t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_286_linordered__field__no__lb,axiom,
! [X4: real] :
? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).
% linordered_field_no_lb
thf(fact_287_linordered__field__no__ub,axiom,
! [X4: real] :
? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_288_inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_289_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_290_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_291_nonzero__imp__inverse__nonzero,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ A )
!= zero_zero_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_292_nonzero__inverse__inverse__eq,axiom,
! [A: real] :
( ( A != zero_zero_real )
=> ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_293_nonzero__inverse__eq__imp__eq,axiom,
! [A: real,B: real] :
( ( ( inverse_inverse_real @ A )
= ( inverse_inverse_real @ B ) )
=> ( ( A != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_294_inverse__zero__imp__zero,axiom,
! [A: real] :
( ( ( inverse_inverse_real @ A )
= zero_zero_real )
=> ( A = zero_zero_real ) ) ).
% inverse_zero_imp_zero
thf(fact_295_field__class_Ofield__inverse__zero,axiom,
( ( inverse_inverse_real @ zero_zero_real )
= zero_zero_real ) ).
% field_class.field_inverse_zero
thf(fact_296_divide__le__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% divide_le_0_iff
thf(fact_297_divide__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_right_mono
thf(fact_298_zero__le__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% zero_le_divide_iff
thf(fact_299_divide__nonneg__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_300_divide__nonneg__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonneg_nonpos
thf(fact_301_divide__nonpos__nonneg,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y )
=> ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_nonpos_nonneg
thf(fact_302_divide__nonpos__nonpos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_303_divide__right__mono__neg,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% divide_right_mono_neg
thf(fact_304_divide__neg__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_neg_neg
thf(fact_305_divide__neg__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_neg_pos
thf(fact_306_divide__pos__neg,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ Y @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).
% divide_pos_neg
thf(fact_307_divide__pos__pos,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y )
=> ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).
% divide_pos_pos
thf(fact_308_divide__less__0__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% divide_less_0_iff
thf(fact_309_divide__less__cancel,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ B @ A ) )
& ( C != zero_zero_real ) ) ) ).
% divide_less_cancel
thf(fact_310_zero__less__divide__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_divide_iff
thf(fact_311_divide__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono
thf(fact_312_divide__strict__right__mono__neg,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_real @ B @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_313_inverse__less__imp__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ B @ A ) ) ) ).
% inverse_less_imp_less
thf(fact_314_less__imp__inverse__less,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).
% less_imp_inverse_less
thf(fact_315_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide_nat @ M @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_316_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_317_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_318_complete__real,axiom,
! [S2: set_real] :
( ? [X4: real] : ( member_real @ X4 @ S2 )
=> ( ? [Z: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ? [Y4: real] :
( ! [X4: real] :
( ( member_real @ X4 @ S2 )
=> ( ord_less_eq_real @ X4 @ Y4 ) )
& ! [Z: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z ) )
=> ( ord_less_eq_real @ Y4 @ Z ) ) ) ) ) ).
% complete_real
thf(fact_319_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_320_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_321_div__neg__pos__less0,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_322_neg__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_323_pos__imp__zdiv__neg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_324_zdiv__int,axiom,
! [A: nat,B: nat] :
( ( semiri2019852685at_int @ ( divide_divide_nat @ A @ B ) )
= ( divide_divide_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B ) ) ) ).
% zdiv_int
thf(fact_325_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_326_zdiv__mono2,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_327_zdiv__eq__0__iff,axiom,
! [I2: int,K: int] :
( ( ( divide_divide_int @ I2 @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I2 )
& ( ord_less_int @ I2 @ K ) )
| ( ( ord_less_eq_int @ I2 @ zero_zero_int )
& ( ord_less_int @ K @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_328_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_329_zdiv__mono2__neg,axiom,
! [A: int,B2: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ B2 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_330_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_331_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq_int @ L @ K )
=> ( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_332_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_333_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_334_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I2: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
= ( ord_less_eq_int @ K @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_335_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_336_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N )
= A )
& ! [Y2: real] :
( ( ( ord_less_real @ zero_zero_real @ Y2 )
& ( ( power_power_real @ Y2 @ N )
= A ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_337_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R: real] :
( ( ord_less_real @ zero_zero_real @ R )
& ( ( power_power_real @ R @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_338_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_339_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri2019852685at_int @ A3 ) @ ( semiri2019852685at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_340_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_341_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri2019852685at_int @ A ) ) )
& ( ~ P
=> ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri2019852685at_int @ B ) ) ) ) ).
% int_if
thf(fact_342_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A3: nat,B3: nat] :
( ( semiri2019852685at_int @ A3 )
= ( semiri2019852685at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_343_int__ops_I1_J,axiom,
( ( semiri2019852685at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_344_div__gr__imp__gr__divisor,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less_nat @ X @ ( divide_divide_nat @ N @ M ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% div_gr_imp_gr_divisor
thf(fact_345_div__eq__0__conv,axiom,
! [N: nat,M: nat] :
( ( ( divide_divide_nat @ N @ M )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( ord_less_nat @ N @ M ) ) ) ).
% div_eq_0_conv
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( listSl97544552lice_a @ nil_a @ k )
= nil_list_a ) ).
%------------------------------------------------------------------------------